Problem: Let $f(x) = 6x^{2}-9x-4$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $6x^{2}-9x-4 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 6, b = -9, c = -4$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 6 \cdot -4}}{2 \cdot 6}$ $ x = \dfrac{9 \pm \sqrt{177}}{12}$ $ x = \dfrac{9 \pm \sqrt{177}}{12}$ $x =\dfrac{9 \pm \sqrt{177}}{12}$